Trigonometric Equations (Cambridge (CIE) AS Maths: Pure 1): Exam Questions

Find all solutions to the equation cos in the interval -2 ≤ ≤ 2, giving your answers in radians as multiples of .

Find all solutions to the equation 5 sin 3x = 1 in the interval 0 ≤ x ≤ , giving your answer in radians to three significant figures.

Show that the equation 2 sin² x + 3 cos x = 0 can be written in the form

acos²x+bcosx+c=0, where a, b and c are integers to be found.

Hence, or otherwise, solve the equation 2 sin² x + 3 cos x = 0 for -180° ≤ × ≤ 180°

Given that sin = find the possible values of cos and tan .

Solve the equation 2 sin 2 = 1 for 0 ≤ ≤ 2

Solve the equation 2 sin x =

A right-angled triangle has hypotenuse 8cm. One of its other sides is 5cm.

Find exact values for sin , cos and tan , where is the smallest angle in the triangle.

Solve the equation 2 sin x cos x = cos x for

Show that (x + 1)(x-2)(x-3)x³ - 4x² + x + 6.

Hence, or otherwise, solve the equation tan³x - 4 tan² x + tan x + 6 = 0 for

0° ≤ x ≤ 360°, giving your answers to 1 decimal place where appropriate.

A seagull sits on the surface of the sea and moves up and down as waves pass.

Its height, h metres, above its position in calm water is modelled by the function h= sin(180t) where t is the time in seconds after timing commences.

Sketch a graph of h against t for 0 ≤ t ≤ 10 showing the coordinates of the points of intersection with the t axis.

How many times in the first minute after timing commences is the seagull 0.25 metres above its calm water position?

Find the time at which the seagull is first 0.25m above its calm water position and moving downwards. Give your answer to 3 significant figures.

Solve the equation 2 sin = 3 cos for 0 ≤ ≤ 2, giving your answers to 3 significant figures.

Solve the equation 2 sin² = cos + 1 for -180° ≤ ≤ 180°.

Given that the angle is obtuse and that sin , find the exact value of cos .

Solve the equation tan 2x = for -180° ≤ x ≤ 180°

Solve the equation 2 tan x - sin x = 0 for ≤ x ≤

An isosceles triangle has sides 8 cm, 8 cm and 4 cm and equal base angles .

Find exact values for sin , cos and tan .

Find all the solutions to the equation in the interval giving your answers in radians as multiples of .

Find all the solutions to the equation 6 sin² x + 7 sinx - 3 = 0 in the interval, giving your answers in radians to three significant figures.

Show that satisfies the equation 8x³ - 4x² - 6x + 3 = 0.

Hence solve the equation 8 cos³ x - 4 cos²x - 6 cos x+ 3 =0 for 0° ≤ × ≤ 360°

A seagull sits on the surface of the sea and moves up and down as waves pass.

Its height, h metres, above its position in calm water is modelled by the functionwhere Chis the time in seconds afer tming commenced.

Find the first time the seagull is 0.3 metres above its calm water position.

Give your answer to 2 decimal places.

How many times in the first minute after timing commences is the seagull 0.3 metres above its calm water position?

Solve the equation 3 sin 3 = 4 cos 3 in the interval 0 ≤ ≤ , giving your answers to 3 significant figures.

Solve the equation 6 cos² = sin 2 + 5 for -180° ≤ ≤ 180°, giving your answers to 1 decimal place where appropriate.

Given that the angle is reflex and that cos = , find the exact value of tan .

Solve the equation 2 sin² 3x = 1 for

Solve the equation 3 sin(2x + 30°) = tan(2x + 30º) for - 180° ≤ x ≤ 180, giving your answers to 1 decimal place where appropriate.

For the triangle in the diagram find exact values for sin x, cos x and tan x.

Find all the values of x in the range 0° ≤ x ≤ 180° which satisfy the equation

6 tan³ 2x - 7 tan² 2x - tan 2x + 2 = 0, giving your answers to 1 decimal place.

Find all the solutions to the equation 2 cos 2 = 4 sin 2 cos 2 in the interval

0 ≤ ≤ 2, giving your answers in radians as multiples of .

Find all the solutions to the equation 3 cos²4x + 13 cos 4x - 10 = 0 in the interval

, giving your answers in radians to three significant figures.

A seagull sits on the surface of the sea and moves up and down as waves pass.

Its height, h metres, above its position in calm water is modelled by the function where t is the time in seconds after timing commences.

Find the amount of time the seagull is more than 0.5 metres above its calm water position in the first 20 seconds after timing commences.

Give your answer correct to 3 significant figures.